Question: Simplify and expand the following expression: $ \dfrac{2p}{p - 10}+\dfrac{p - 7}{2p - 2} $
In order to add expressions, they must have a common denominator. Get both fractions over a common denominator of $(p - 10)(2p - 2)$ Multiply the first term by $\dfrac{2p - 2}{2p - 2}$ $ \begin{align*} \dfrac{2p}{p - 10} \times \dfrac{2p - 2}{2p - 2} & = \dfrac{(2p)(2p - 2)}{(p - 10)(2p - 2)} \\ & = \dfrac{4p^2 - 4p}{(p - 10)(2p - 2)}\end{align*} $ Multiply the second term by $\dfrac{p - 10}{p - 10}$ $ \begin{align*} \dfrac{p - 7}{2p - 2} \times \dfrac{p - 10}{p - 10} & = \dfrac{(p - 7)(p - 10)}{(2p - 2)(p - 10)} \\ & = \dfrac{p^2 - 17p + 70}{(2p - 2)(p - 10)}\end{align*} $ Now we have: $ = \dfrac{4p^2 - 4p}{(p - 10)(2p - 2)} + \dfrac{p^2 - 17p + 70}{(2p - 2)(p - 10)} $ Now both terms have a common denominator we can simply add the numerators: $ = \dfrac{4p^2 - 4p + p^2 - 17p + 70}{(p - 10)(2p - 2)} $ $ = \dfrac{5p^2 - 21p + 70}{(p - 10)(2p - 2)}$ Expand the denominator: $ = \dfrac{5p^2 - 21p + 70}{2p^2 - 22p + 20}$